Confidence Sets for Statistical Classification (II): Exact Confidence Sets

Classification has applications in a wide range of fields including medicine, engineering, computer science and social sciences among others. Liu et al. (2019) proposed a confidence-set-based classifier that classifies a future object into a single class only when there is enough evidence to warrant this, and into several classes otherwise. By allowing classification of an object into possibly more than one class, this classifier guarantees a pre-specified proportion of correct classification among all future objects. However, the classifier uses a conservative critical constant. In this paper, we show how to determine the exact critical constant in applications where prior knowledge about the proportions of the future objects from each class is available. As the exact critical constant is smaller than the conservative critical constant given by Liu et al. (2019), the classifier using the exact critical constant is better than the classifier by Liu et al. (2019) as expected. An example is provided to illustrate the method.

An Exploration Toward a Unified Failure Criterion

For components with different defects, selecting a proper criterion to predict their failure is very important, but sometimes this brings confusion to engineers. In this paper, we explore to establish a unified failure criterion for defects with various geometries. First, a fundamental and universal law on failure that all criteria should follow, so-called the zeroth law of failure, is introduced, and the failure is completely governed by the local status of failure determining zone (FDZ), such as the stress distribution, material properties, and geometrical features. Failure criteria lacking a local dimension parameter within FDZ may have limited applicability, such as the traditional strength and fracture criteria. We choose the blunt V-notch as an example to demonstrate how to establish a unified failure criterion for quasi-brittle materials, and a series of experiments are carried out to verify its applicability. The proposed unified failure criterion and some existing failure criteria are also discussed and compared. The failure criteria that only include a single critical constant are incapable of reflecting the whole stress field information and local geometrical features of the FDZ. Our proposed unified failure criterion is expressed with a two-parameter function and has a wider applicability.

Positivity of the two-dimensional Brown—Ravenhall operator

We determine the critical coupling of the two-dimensional Brown–Ravenhall operator with Coulomb potential. Boundedness from below has essentially been proven by Bouzouina. However, that work contains a trivial error leading to an incorrect constant that is exactly half of the actual critical constant. Furthermore, we show that the operator is in fact positive. Our proof of that is, for the most part, analogous to Tix's proof of the corresponding result for the three-dimensional operator.

A New Approach to Estimate the Critical Constant of Selection Procedures

A solution to the ranking and selection problem of determining a subset of size containing at least of the best from normal distributions has been developed. The best distributions are those having, for example, (i) the smallest means, or (ii) the smallest variances. This paper reviews various applicable algorithms and supplies the operating constants needed to apply these solutions. The constants are computed using a histogram approximation algorithm and Monte Carlo integration.

A critical oscillation constant as a variable of time scales for half-linear dynamic equations

We present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y Δ))Δ+p(t)Φ(y σ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.

A critical constant for the k nearest-neighbour model

Let 𝒫 be a Poisson process of intensity 1 in a square S n of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph G n,k . We prove that there exists a critical constant c crit such that, for c < c crit, G n,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > c crit, G n,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

A critical constant for the k nearest-neighbour model

Let 𝒫 be a Poisson process of intensity 1 in a square Sn of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that, for c < ccrit, Gn,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > ccrit, Gn,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

On normality assumptions for claims in insurance

The aim of this paper is to discuss effects of deviations from hypothetized normality. Two models are considered, one is the first pension pillar (and we consider here very small samples, which plays some role at start of some pension system or at early phases of it) and second one of modeling for IBNR (here we consider mid-samples). We will show that at early phases of 1stpension pillar in Slovakia the estimation of upper probability of oversizing of critical constant given by Potocký and Stehlík, 2005, fits well. For the case of IBNR reserves, the date given by Stelljes, 2006, are significantly more skewed and thus further research is needed for appropriate modelling of these reserves.